Is it possible for a Roku stick to short out the HDMI port on the TV?

There are a number of things that can cause issues with HDMI ports.  Repeated cable insertion and removal can cause bent or broken pins in the port or the device/HDMI cable, physical damage to the cable or ports from being bumped or dropped, dust accumulation within the HDMI port, poor quality HDMI cables or cables that use an old protocol, or even excessive force when plugging the HDMI cable/device into the port.  Electrical surges or even static electricity can damage the sensitive electronics inside the port.  It is possible but not likely that a device like a Roku can create an issue with a power spike.  Damage by a Roku to an HDMI port is more likely caused by improperly inserting or removing the device from the port than by any surge caused by it.

If not caused by physical damage to the ports, then it is likely that a surge coming from the TV or the outlet the TV damaged the control board that controls the HDMI ports.  There are a lot of factors that can contribute to that, including the age and brand of the TV and quality of components they used for that model, the wear and tear on the ports themselves with devices being used/inserted/removed in them, how clean your electricity is in your area, in terms of power spikes/surges, and whether you use a surge protector and/or home house surge to protect from those spikes.  

My area here gets a number of short outages, 2 seconds or less, that causes a number of surges, which has over time damaged components in a home receiver and fridge I have, even with surge protecters and a whole house surge as they do not provide 100 percent foolproof protection.  I have added surges that keep the power off for 4 minutes after an outage to protect from those little spikes from damaging the components, which adds a layer of protection to those sensitive components.

It sounds like your parents' TV had a board that was starting to fail and then failed completely after trying to use the other ports. 

You could check to see if there is an issue with the Roku by plugging it into a different TV that you know has working HDMI ports and see if the Roku device itself is failing or has an issue as they like any other electronic device can fail due to power spikes or other electronic failures.   Or you can try plugging a different HDMI device (such as a Blu-ray player or cable box) into the TV and see if those ports work with that device.

I had a Roku stick do the same to my Samsung TV as well a few years back. While I agree a R-Stick should not be able to have that effect on the input of an HDMI port, this stick broke a few HDMI ports in the same way... I had 2 of 3 ports I tested on that TV, both of which we fried by the Roku stick and I tried another TV (because I was gluten for punishment and the same happened.) It was using the power adapter provided with the Roku stick. I even tried another equal voltage/amperage output known good working power supply and no change. Wish I held on to it to pick it apart and see what the cause was but I was so disgusted I threw it in the trash.

It's not unheard of for hardware to turn faulty (or be open to a hack) but I learned my lesson that the last thing I'll ever buy is another Roku. Nvidia Shield has been pretty reliable and a fantastic option. I also used Google's Streaming puck and a security hardened Raspberry Pi for a Plex Streaming my home library. It's a shame because I liked Roku at first but after that experience, never again!